Algebraic K-theory and Motivic Cohomology

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Weibel proposes to do research in the related fields of algebraic K-theory and motivic theory. First, he proposes to complete the proof of a theorem which describes the relationship that K-theory and motivic cohomology have with the related fields of algebraic geometry and number theory. A second part of the project is to apply new motivic techniques to long-standing problems in algebraic geometry dealing with singularities. The third part of the project will be to study the internal structure of motives in an attempt to explain the behavior of Chow groups, which are classical invariants of algebraic varieties.Specifically, Weibel proposes three strands of research. He will try to establish the existence of Rost varieties, completing the verification of the Bloch-Kato Conjecture. This conjecture links K-theory to number theory and etale cohomology. He will also study the relationship between the singularities of a variety, its K-theory and its cdh-cohomology, using recently developed cohomological techniques. This involves several classical questions about the K-theory of rings of integers in global fields which have been a focal point of research in K-theory for the last 35 years, especially in the last decade. In addition, he will study the structure of Motives, especially the structure of slice filtration, as a way of attacking conjectures about vanishing of Chow groups and higher Chow groups.